xd] of polynomials in d real variables with real coefficients, provided with the LF-topology τind introduced in Example I in Section 1.3 of the lecture notes

Universität Konstanz

Fachbereich Mathematik und Statistik Dr. Maria Infusino

Patrick Michalski

TOPOLOGICAL VECTOR SPACES II–WS 2019/2020 Exercise Sheet 3

This exercise sheet aims to assess your progress and to explicitly work out more details of some of the results proposed in the previous lectures. Please, hand in your solutions in postbox 18 near F411 by Wednesday the 11th of December at 13:30. The solutions to this assignment will be discussed in the tutorial on Tuesday the 17th of December (17:00–18:30) in P812.

1) Consider the space R[x1, . . . , xd] of polynomials in d real variables with real coefficients, provided with the LF-topology τ_ind introduced in Example I in Section 1.3 of the lecture notes. Prove the following two facts:

a) The LF-topologyτindonR[x1, . . . , xd]is the finest locally convex topology on this space.

b) Every linear map from(R[x₁, . . . , x_d], τ_ind)into any t.v.s. is continuous.

2) Consider the space C_c^∞(Ω) (with Ω ⊆ R^d open) of test functions provided with the LF- topology τind introduced in Example II in Section 1.3 of the lecture notes. Show that (C_c^∞(Ω), τ_ind)is not metrizable.

3) LetE be a vector space overKendowed with the projective topologyτ_proj w.r.t. the family {(E_α, τα, fα) : α ∈A}, where each (Eα, τα) is a locally convex t.v.s. over K and each fα is a linear mapping fromE to Eα. Let (F, τ) be an arbitrary t.v.s. and u be a linear mapping fromF intoE.

a) Show that the mappingu :F → E is continuous if and only if, for each α∈A,fα◦u: F →Eα is continuous.

b) Does the previous statement still hold if we ignore the vector space structures, that is, if we just assume thatE is a set, all (E_α, τ_α) and (F, τ) are topological spaces, each f_α is a mapping fromE to Eα and τproj is the coarsest topology onE such that all mappings fα are continuous?

4) Let (A,≤) be a directed partially ordered set and let (E, τproj) be the projective limit of the family {(E_α, τ_α), α ∈ A} of l.c. t.v.s. w.r.t. the maps {g_αβ : α, β ∈ A, α ≤ β} and {f_α:α∈A}. Show that:

a) gαγ =gαβ◦gβγ for all α≤β ≤γ inA.

b) f_α=g_αβ◦f_β for all α≤β inA.

c) If(F, τ) is another locally convex t.v.s. and for eachα∈A,g_α:F →E_α is a continuous linear map such thatgα=gαβ◦g_βfor allα≤βinA, then there exists a unique continuous and linear mapg:F →E such thatg_α =f_α◦g for all α∈A.